Mixed convection flow past a vertical plate: Stability analysis ...staff.bath.ac.uk/ensdasr/PAPERS/MISCELLANEOUS/IJTS_paper...Mixed convection ﬂow over a vertical ﬂat plate. So

International Journal of Thermal Sciences 48 (2009) 461–474www.elsevier.com/locate/ijts

Mixed convection flow past a vertical plate: Stability analysisand its direct simulation

K. Venkatasubbaiah a, T.K. Sengupta b,∗

a Department of Aerospace Engineering, IIT Kanpur, Currently at Amrita School of Engineering, Coimbatore, Indiab Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India

Received 18 March 2008; accepted 18 March 2008

Available online 25 April 2008

Abstract

Stability of mixed convection flow past vertical flat plate have been investigated here using compound matrix method (CMM) to solve linearizedequations arising out of a spatial analysis and by direct numerical simulation (DNS) using a Boussinesq approximation. CMM is applied to studyspatial stability of mixed convection flow past a vertical plate. A double loop in the neutral curve is shown in the forced convection limit ofopposing flows for the first time. To check this and other observations of the linear analysis, two-dimensional Navier–Stokes equations are alsosolved here with the buoyancy term represented by the Boussinesq approximation. This type of receptivity study, by direct simulation havebeen extended here for mixed convection flows. Also, we establish the presence of a spatio-temporally growing wave-front for this flow—thatwas shown to exist in Sengupta et al. [T.K. Sengupta, A.K. Rao, K. Venkatasubbaiah, Spatio-temporal growing wave-fronts in spatially stableboundary layers, Phys. Rev. Lett. 96 (224504) (2006) 1–4; T.K. Sengupta, A.K.Rao, K. Venkatasubbaiah, Spatio-temporal growth of disturbancesin a boundary layer and energy based receptivity analysis, Phys. of Fluids 18 (094101) (2006) 1–9] for boundary layer developing over a horizontalflat plate in the absence of heat transfer.© 2008 Elsevier Masson SAS. All rights reserved.

Keywords: Mixed convection flow; Linear and nonlinear stability; Nonlinear receptivity; Direct numerical simulation; Compound matrix method

1. Introduction

Despite significant advances made in hydrodynamic stabilitytheory, there are many issues of flow transition (those affectedby more than one physical mechanisms or due to the pres-ence of simultaneous multiple unstable modes), remain incom-pletely understood. Mixed convection flow is a typical exam-ple, seen in many engineering devices. In pure hydrodynamicscenario, flow instabilities are caused due to complex interac-tions between inertial and viscous mechanisms of exchangingmomentum-detailed description of the same are given in Drazinand Reid [3]. When heat transfer effects are included, ensuinginstability requires taking into consideration of energy transfer,along with the added buoyancy-induced effects in momentum

* Corresponding author.E-mail addresses: [email protected]

(K. Venkatasubbaiah), [email protected] (T.K. Sengupta).

1290-0729/$ – see front matter © 2008 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.ijthermalsci.2008.03.019

conservation equation. These added effects make the corre-sponding flow instability studies further complicated. A com-prehensive review of heat transfer aspect of mixed convectionare given in Gebhart et al. [4]. It has been noted by Brewster andGebhart [5] that in the mixed convection regime, instability isdue to growth of small disturbances that can be studied by lin-earized governing equation. Such an approach for instability offlows over a horizontal plate with heat transfer have been stud-ied in Wu and Cheng [6], Chen and Mucoglu [7], Sengupta andVenkatasubbaiah [8] and other references contained therein. InSengupta and Venkatasubbaiah [8], it has been shown that thereexist a critical buoyancy parameter, above which a very unsta-ble higher frequency variation ensues, in addition to the purehydrodynamic mode characterized by lower frequency distur-bances. Both of these modes for the flow past heated horizontalplates were obtained by the linearized spatial stability analysisperformed using CMM. For the first time, existence of criticalbuoyancy parameter was shown theoretically that has been de-tected in the experiments by Wang [9] earlier.

462 K. Venkatasubbaiah, T.K. Sengupta / International Journal of Thermal Sciences 48 (2009) 461–474

Nomenclature

Re . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reynolds numberGr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grashof numberPr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prandtl numberRix . . . . . . . . . . . . . . . . . . . . . . . . . . Buoyancy parameterδ∗ . . . . . . . . . . . . . . . . . . . . . . . . displacement thickness

k . . . . . . . . . . . . . . . . . non-dimensional wave numberβ0 . . . . . . . . . . . . . non-dimensional circular frequencyψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stream functionω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vorticityFf . . . . . . . . . . . non-dimensional frequency parameter

For flow past vertical plates, induced body force due to heattransfer is either parallel or anti-parallel to the mean convectiondirection—as shown schematically in Fig. 1. These are com-monly referred to as assisting and opposing flows. Like otherconvection dominated unseparated flows, instability of mixedconvection flow past vertical plates also occur via growth ofsmall disturbances. Thus, this has been studied by linear anal-ysis in Mucoglu and Chen [10], Brewstar and Gebhart [5] andMoresco and Healey [11]. These analyses have been tradition-ally performed using temporal theory (see, e.g., Mucoglu andChen [10]). The equilibrium assisting flow was obtained by lo-cal non-similarity method and it was noted that the buoyancyforce stabilizes the flow. However, experimental studies showthe instability to be related to spatial growth of disturbanceswhen the flow is excited by fixed frequency sources. Hence aspatial theory is preferred to study the stability of mixed con-vection flows, as in Lee et al. [12]. They reported results for thebuoyancy parameter Rix = Grx/Re2

x in the range between zeroand infinity, where Grx and Rex are the Grashof and Reynoldsnumbers based on current length. Lee et al. [12] and Morescoand Healey [11] have studied mixed convection flow over theentire range of Rix , but reported two unstable modes of dis-turbance for natural convection dominated flows only. In thepresent work, existence of two unstable modes and thereforetwo loops of neutral curve is established in the forced convec-tion regime.

Additionally, in the present study we focus upon both thelinear and nonlinear route of instability to fixed frequency wallexcitations. First, the linear spatial stability results for mixedconvection flow past vertical plate are reported using CMM tosolve the stability equations. CMM allows circumventing thestiffness that is inherent with viscous instability problems. De-tailed description of CMM is to be found in Drazin and Reid[3], Allen and Bridges [13,14] and Sengupta and Venkatasubba-iah [8] and its use for mixed convection flow past vertical plateis reported here for the first time. All these stability analysessuffer from the restriction of either nonlinearity or nonparal-lelism of the mean flow or due to both.

Qualitative and quantitative differences exist between thelinear spatial theory and experimental results for flows withand without heat transfer. For flows without heat transfer, Faseland Konzelmann [15] have studied effects based on solution ofcomplete Navier–Stokes equation to include all possible non-parallel and nonlinear effects. This represents direct numericalsimulation (DNS) of the laboratory experiments of Schubauerand Skramstad [16], where the response of the boundary layerto a vibrating ribbon at different frequencies were investigated.

Fig. 1. Mixed convection flow over a vertical flat plate.

So far, no such studies have been undertaken for mixed convec-tion flows. In the present exercise we also report the completenonlinear, nonparallel analysis of mixed convection flow pastvertical plates by solving the time-dependent Navier–Stokesequation. In flows with heat transfer, Lee et al. [17] have re-ported nonparallel instability studies for flow over inclinedheated platefrom vertical to horizontal position. They used anonparallel formulation based on an order-of-magnitude analy-sis. However, it is necessary to evaluate the results of the paperclosely. For example, the critical Reynolds number (Recr) valueattributed to experimental results in Schubauer and Skramstad[16] is erroneously given as Recr = 378, whereas, Fig. 11 ofSchubauer and Skramstad [16] clearly shows this to be in ex-cess of 400. Lee et al. [17] have calculated this as Recr = 374and claimed excellent agreement with experiment. They alsoobtained this Recr for Ff = β0ν/U2∞ = 63.61 × 10−6, thatis more than six times lower than the actual value given inSchubauer and Skramstad [16]. Here, βo is the circular fre-quency of wall-excitation; ν is the kinematic viscosity and U∞is the boundary layer edge velocity. In a critique of early non-parallel methods, Tumin [18] cites neglecting the dependenceof eigen functions on streamwise coordinate as responsible forthe discrepancy. He used the theory of multiple scales to incor-

K. Venkatasubbaiah, T.K. Sengupta / International Journal of Thermal Sciences 48 (2009) 461–474 463

porate nonparallel effects for both traveling wave and stationarylongitudinal-vortices mode solutions. Stabilizing effects due tononparallelism of the mean flow was reported by Tumin [18].Thus, the past studies have not been able to provide correct Recr

at the correct Ff , based on linearized quasi-parallel and weaklynonparallel models for the stability properties of mixed convec-tion flows.

Present study attempts to investigate the linearized parallelspatial stability of the mixed convection assisting and oppos-ing flows in the forced convection limit by using CMM and tostudy nonparallel, nonlinear stability of the same based on thesolution of complete Navier–Stokes and energy equations to in-clude all possible effects for two-dimensional flows. We haveused the wall excitation model of Fasel and Konzelmann [15]for the DNS of mixed convection flow over the vertical plate.

The paper is structured in the following manner. Governingequations are given in the next section. Linear stability analysisis discussed in Section 3. Direct numerical simulation of theflow is reported in Section 4. The paper closes with a summaryand some comments in Section 5.

2. The governing equations

We consider the laminar two-dimensional motion of fluidpast a semi-infinite vertical plate, with the free stream velocityand temperature denoted by, U∞ and T∞. The flow configu-rations for assisting and opposing flows are shown in Fig. 1.We consider the flow over isothermal vertical plate, for whichthe surface temperature (Tw) is greater than the free streamtemperature (T∞). Governing equations are presented in non-dimensional form with the buoyancy term modeled by theBoussinesq approximation. Non-dimensionalization of equa-tions are performed by introducing an appropriate length (L),velocity (U∞), temperature (�T = Tw − T∞) and pressurescales (ρU2∞). These equations for the velocity and tempera-ture fields are as given in Gebhart et al. [4],

∇ · �V = 0 (1)

D �VDt

= ± Gr

Re2T − ∇p + 1

Re∇2 �V (2)

DT

Dt= 1

Re Pr∇2T (3)

where T = (T ∗ − T∞)/�T and Gr = gβt�T L3/ν2; Re =U∞L/ν and Pr = ν/α where α is the thermal diffusivity ofthe fluid; T ∗ is the dimensional temperature in the field. In themomentum conservation equation, the quantity Gr/Re2, is alsoknown as the Richardson number (Ri). Positive and negativesigns of Ri refer to assisting and opposing flows, respectively.The Grashof number weighs the relative importance of buoy-ancy and viscous diffusion terms and in the mixed convectionregime, Ri is of order one.

3. Linear stability analysis

For studying linear instability of mixed convection flow overa vertical flat plate, equilibrium and disturbance fields are sep-

arated and their equations are similar to that obtained in Sen-gupta and Venkatasubbaiah [8] and only essential details aregiven below.

3.1. Equilibrium or mean flow equations

The mean flow equations are obtained by invoking boundarylayer approximation for two-dimensional steady incompress-ible flow with constant properties and Boussinesq approxima-tion. The mean flow equations are obtained using the follow-ing variables: ηs = y

√U∞/νx for the independent variable;

u/U∞ = F ′ and (T ∗ − T∞)/(Tw − T∞) = T for the dependentvariables and are obtained from the solution of (Oosthuizen andNaylor [19]),

F ′′′ + FF ′′

2± RixT = 0 (4)

T ′′ + Pr

2FT ′ = 0 (5)

where Rix = Grx/Re2x is the buoyancy parameter and ± signs

are for the buoyancy term for assisting and opposing flows, re-spectively. In these equations, primes indicate derivatives withrespect to ηs . As Rix is a function of x, similarity solution doesnot exist. However, if Rix is small, then one can obtain the per-turbation solution by representing the dependent variables by,

F = F0 + RixF1 + (Rix)2F2 + · · ·

T = θ0 + Rixθ1 + (Rix)2θ2 + · · ·

where F0 and θ0 are the solution for pure forced convectioncases for Rix = 0. Thus, a formal perturbation series analysisprovides the governing equations for the mean flow as givenby the following sets of equations—expressed for up to secondorder only,

F ′′′0 + F0F

′′0

2= 0 (6)

F ′′′1 + F0F

′′1

2+ F1F

′′0

2± θ0 = 0 (7)

F ′′′2 + F0F

′′2

2+ F1F

′′1

2+ F2F

′′0

2± θ1 = 0 (8)

θ ′′0 + Pr

2F0θ

′0 = 0 (9)

θ ′′1 + Pr

2F0θ

′1 + Pr

2F1θ

′0 = 0 (10)

θ ′′2 + Pr

2F0θ

′2 + Pr

2F1θ

′1 + Pr

2F2θ

′0 = 0 (11)

These equations are solved subject to the boundary conditionsat ηs = 0: F0 = F ′

0 = F1 = F ′1 = F2 = F ′

2 = 0 and θ0 = 1, θ1 =θ2 = 0 and as ηs → ∞: F ′

0 = 1, F ′1 = F ′

2 = 0 and θ0 = θ1 =θ2 = 0.

The mean flow in air is obtained by solving Eqs. (6) to (11)by the four-stage Runge–Kutta method using shooting tech-nique with Pr = 0.7, and by taking maximum co-ordinate,(ηs)max = 12 divided into 4000 equal sub-intervals. For differ-ent Re and Rix , mean flow solutions are obtained for assistingand opposing flows. Non-dimensional velocity and temperature


Fig. 2. Mean flow profiles for assisting ( with Rix = 0.1) and opposing (withRix = 0.001) flows given by (a) first order perturbation solution for assistingflow; (b) second order perturbation solution for assisting flow and (c) first orderperturbation solution for opposing flow.

profiles are as shown in Fig. 2, obtained up to second order ac-curacy, for the assisting (with Rix = 0.1) and opposing flows(with Rix = 0.001). There is only a small deviation between thefirst and second order quantities for the velocity field for theassisting flow at the high value of Rix = 0.1. Stability studiesconducted here are for lower Rix values, for which higher or-der effects will be further negligible. We will also show that thestability of the two mean flows of a and b in Fig. 2, have neg-ligible difference for the eigenvalues and the eigen-spectrum.Here, we do not see any difference in the temperature distribu-tion inside the boundary layer, when the second order terms areadded.

3.2. Stability equations

Here, linear stability equations for two-dimensional mixedconvection flow over a vertical plate have been derived by start-ing from the non-dimensional equations (1) to (3). All physicalvariables are split into the mean (given by Eqs. (6) to (11)) andthe disturbance components in the following,

u(x, y, t) = U (x, y) + εud(x, y, t)

v(x, y, t) = V (x, y) + εvd(x, y, t)

p(x, y, t) = P (x, y) + εpd(x, y, t)

T (x, y, t) = T (x, y) + εTd(x, y, t)

with ε as the non-dimensionalizing small parameter of the prob-lem. Stability equations are obtained by making additional par-allel flow assumption (U = U(y), V = 0 and T = T (y)), sothat a normal mode spatial instability analysis is possible bylooking for a solution of the linearized equations of the follow-ing form:

[ud, vd,pd, Td ] = [f (y),φ(y),π(y),h(y)

]ei(kx−β0t) (12)

For the spatial analysis, one fixes a real frequency that is indi-cated here by β0. After substituting Eq. (12) into Eqs. (1) to (3)and simplifying, one obtains the following system of equationsfor disturbance quantities,

i(kU − β0)(k2φ − φ′′) + ikU ′′φ

= ± Gr2ikh′ − 1

(φiv − 2k2φ′′ + k4φ) (13)
Re Re
i(kU − β0)h + T ′φ = 1

Re Pr(h′′ − k2h) (14)

In deriving these equations, length scale (L) is chosen as thedisplacement thickness (δ∗) of the boundary layer. These are thewell-known Orr–Sommerfeld equations for mixed convectionflows, governing the amplitudes of disturbance normal velocityand temperature field. Here, primes denote differentiation withrespect to y. Eqs. (13) and (14) are to be solved subject to thesix boundary conditions:

at y = 0: φ,φ′ = 0 and h = 0 (15)

as y → ∞: φ,φ′, h → 0 (16)

Homogeneous boundary condition for h at the wall correspondsto eigenvalue analysis. In contrast, for receptivity analyses,h = h(y = 0, t) is prescribed at the wall, representing a specificthermal input. As shown earlier in Sengupta et al. [20], thesetwo analyses are related through the disturbance amplitude ex-pression for linear systems. Eqs. (13) and (14), together withthe boundary conditions (15) and (16), reveal that the tempera-ture field, as given by Eq. (14), decouples from the velocity fieldin the free stream (y → ∞), as T ′ ≈ 0 there. Thus, the charac-teristic modes at free stream are given by: λ5,6 = ∓S, whereS = [k2 + iRe Pr(k − β0)]1/2. However, the disturbance mo-mentum equation is not decoupled, as Eq. (13) at the free streamsimplifies (with U = 1 and mean flow derivatives as zero) to,

i(k − β0)(k2φ − φ′′)

= ± Gr

Re2ikh′ − 1

Re(φiv − 2k2φ′′ + k4φ) (17)

This equation for the disturbance amplitude of normal compo-nent of velocity represents a forced dynamics, with the thermalfield providing the forcing. The homogeneous part of the so-lution is governed by the following characteristic exponents,λ1,2 = ∓k and λ3,4 = ∓Q, where Q2 = k2 + iRe(k − β0). Outof these six characteristic values, we discard those modes thatgrow with y in CMM. Thus, the admissible fundamental solu-tion components are given by,

φ1 = e−ky; φ3 = e−Qy and φ5 = e−Sy (18)

when the real part of k,Q and S are all positive. We representthe governing stability equations as a set of six first order ordi-nary differential equations by introducing the vector: u(y, .) =[u1(y, .), u2(y, .), u3(y, .), u4(y, .), u5(y, .), u6(y, .)]T . Where,u1 = φ, u2 = φ′, u3 = φ′′, u4 = φ′′′, u5 = h and u6 = h′. Thegoverning system of equations given by Eqs. (13) and (14) canbe written as,

{u′j } = [A]{uj } (19)

where the non-zero elements of the matrix A are given by,a12 = 1, a23 = 1, a34 = 1, a41 = −a, a43 = b, a46 = c, a56 = 1,a61 = e, a65 = d ; with a = k4 + iRe kU ′′ + iRe k2(kU −β0); b = 2k2 + iRe(kU − β0); c = ±ikGr/Re; d = k2 +iRe Pr(kU − β0) and e = Re PrT ′.

These equations are further modified for CMM, details ofwhich can be obtained in Sengupta and Venkatasubbaiah [8].One obtains the induced system equations for the present prob-lem as,


y′1 = y2 (20)

y′2 = by1 + cy4 + y5 (21)

y′3 = y4 + y6 (22)

y′4 = dy3 + y7 (23)

y′5 = cy7 + y11 (24)

y′6 = y7 + y8 + y12 (25)

y′7 = dy6 + y9 + y13 (26)

y′8 = by6 + y9 − cy10 + y14 (27)

y′9 = by7 + dy8 + y15 (28)

y′10 = y16 (29)

y′11 = −ay1 + cy13 (30)

y′12 = y13 + y14 (31)

y′13 = ey1 + dy12 + y15 (32)

y′14 = ay3 + by12 + y15 − cy16 + y17 (33)

y′15 = ey2 + ay4 + by13 + dy14 + y18 (34)

y′16 = ey3 + y19 (35)

y′17 = ay6 + y18 − cy19 (36)

y′18 = ey5 + ay7 + dy17 (37)

y′19 = ey6 + y20 (38)

y′20 = ey8 − ay10 + by19 (39)

where the primes indicate differentiation with respect to y, thewall-normal coordinate. We note that the usage of asymptoticboundary conditions at y = Y∞ allows us to convert the originalboundary value problem to an initial value problem, with theinitial conditions as given in Sengupta and Venkatasubbaiah [8].

Main aim of the instability study is to identify critical param-eters that mark the onset of instability, i.e. to obtain Recr and thecorresponding circular frequency, βcr . Here we present resultsfor some representative cases that will be used to compare withDNS results. Neutral curves for assisting and opposing flowsare shown in Figs. 3 and 4, respectively. Critical parameters forassisting and opposing flows are tabulated and shown in Ta-bles 1 and 2.

In Fig. 3(a), the neutral curve for the case of Rix = 0.0,i.e. for the Blasius profile is shown—that is computed here tovalidate the present formulation and method with the resultsreported earlier in Sengupta et al. [20]. The essential differ-ence between the two methods lies in the fact that we solvethe complete set of disturbance equation here (both Eqs. (13)and (14)), as compared to that in Sengupta et al. [20], whereonly Eq. (13) was solved by CMM. The calculated eigenvalues(as shown in tabulated form in Sengupta and Venkatasubba-iah [8]) match exactly with that in Sengupta et al. [20]. Also, theneutral curve for the most unstable mode, as shown in Fig. 3(a),gives Recr = 519.018 and βcr = 0.12 that is identical with pre-viously published results. This validates the present formulationand the method of linear stability analysis.

In Fig. 3(b), the neutral curve for the assisting flow withRix = 0.01 is shown, giving Recr = 589.3 and correspondingβcr = 0.122, that indicates stabilization of the assisting flow. In

Fig. 3. Spatial amplification contours shown in (Re − β0) plane for assistingflows for indicated values of Rix . The neutral curves (kimag = 0) are the outer-most contours. Note the rays OA and OB in (a) are chosen for which simulationresults are reported in Fig. 11.

Fig. 3(c), the neutral curve for assisting flow with Rix = 0.1is shown, that indicates further stabilization with Recr = 916.0and βcr = 0.133. This type of alteration of stability propertiesby buoyancy effects have been reported by other investigatorsfor assisting flows. However, quantitative values of growth anddecay rates vary significantly. This can be attributed to differ-ent methods used to solve stability equations. We also note thatfor the case of Fig. 3(c), as the value of Rix = 0.1 is large,one should include second order perturbation terms in the meanflow representation of Eqs. (8) and (11), to investigate the varia-tion of stability properties with retention of perturbation terms.However, when this was performed, no differences for eitherthe eigen-spectrum or the neutral curve was found even for thislarge value of Rix .

In Fig. 4(a), the neutral curve is shown for opposing flowwith Rix = 0.01 that indicates Recr = 440.5 and βcr = 0.123,i.e. a destabilization of the flow in comparison with Rix = 0case. In this figure, we have marked two straight lines originat-ing from the origin and they represent the path taken by constantphysical-frequency disturbances, that will be used for directsimulation of Navier–Stokes and energy equations in the nextsection. When Rix is increased to 0.02, Recr further reduces toRecr1 = 359.5, while βcr1 increases to 0.129—indicating fur-


(a)

(b)

Fig. 4. (a) Spatial amplification contours shown in (Re−β0) plane for opposingflow with Rix = 0.01. The neutral curve (kimag = 0) is indicated by the outercontour. The rays OA and OB are for the constant frequency wall-excitationcases, results of which are shown in Figs. 7(a) and 7(b). The points P and Qcorrespond to the middle of the wall-exciter. (b) Spatial amplification contoursshown in (Re−β0) plane for opposing flow with Rix = 0.04. The neutral curves(kimag = 0) are indicated by the outer contours. The ray from the origin corre-sponds to a constant frequency wall-excitation case result of which is shown inFig. 8. Point P correspond to the middle of the wall-exciter.

Table 1Critical parameters for assisting flows

Case Rix Recr βcr

1 0.0 519.018 0.1202 0.001 526.529 0.1213 0.01 589.30 0.1224 0.1 916.0 0.133

ther destabilization at higher frequencies. Such deteriorationof stability properties for opposing flows have been noted byLee et al. [12] due to buoyancy effects. However, presence oftwo distinct lobes of the neutral curve for opposing flows forRix � 0.02, has not been reported before. Lee et al. [12] in-dicated two maxima for the growth rate versus streamwise dis-tance plot, without distinct lobes in the neutral curve—as shown

Table 2Critical parameters for opposing flows

Case Rix Recr1 Recr2 βcr1 βcr2

1 0.0001 518.2 – 0.120 –2 0.001 511.3 – 0.120 –3 0.01 440.5 – 0.123 –4 0.02 359.5 7705.3 0.129 0.07995 0.03 283.6 3747.4 0.137 0.1186 0.035 248.2 2714.8 0.145 0.1397 0.04 215.29 1996.0 0.154 0.165

here. According to them, the two maxima are due to the pres-ence of two distinct unstable modes. Presence of two lobes ofthe neutral curve here implies presence of two Recr and βcr . ForRix = 0.02, we have already noted one such pair, the other pairis given by, Recr2 = 7705.3 and βcr2 = 0.0799. These criticalparameters for opposing flow are given in Table 2.

Two lobes of neutral curve in Fig. 4(b) for Rix = 0.04case give two critical Reynolds numbers Recr1 = 215.29 andRecr2 = 1996.0, while the corresponding critical circular fre-quencies are βcr1 = 0.154 and βcr2 = 0.165. We note that withincreasing value of Rix , both the critical Reynolds numbersdecrease continuously, with the second one decreasing rathersharply. From Table 2, we note that βcr1 increases from 0.120to 0.154 for 0.0001 � Rix � 0.04 and βcr2 increases from0.0799 to 0.165 for 0.02 � Rix � 0.04. Presence of multi-lobe neutral curves have also been shown for mixed convec-tion flow over horizontal plate in Sengupta and Venkatasubba-iah [8]. The common features of multiple lobes in the neutralcurves for mixed convection flows past horizontal and verti-cal plates at low speeds are due to higher order of the sys-tem, caused by the coupling of momentum and energy equa-tions.

4. Direct numerical simulation

Stability results presented in the previous section are forparallel boundary layer developing over a vertical plate, ob-tained after linearizing the governing equations. In the process,we report the presence of distinct two-lobed neutral curve foropposing flows for the most unstable mode. It is pertinent toinvestigate this further, without being restricted by linear andparallel flow approximations. Also, all linear stability analysissuffers from the normal mode approach, i.e. the various modespresent are sought separately and their actions are consideredto be independent of each other. This can be rectified by the re-ceptivity approach—as in Sengupta et al. [20] for the linearizedresponse or by solving the Navier–Stokes equation with respectto specific excitation. In the following, we follow the latterby performing a direct simulation of the 2D flow field. TheNavier–Stokes equations for incompressible flows are solvedin stream function (ψ ) and vorticity (ω) formulation. The gov-erning equations (1) to (3) are written in (ψ − ω) formulationin an orthogonally transformed plane as given by,

∂(

h2 ∂ψ)

+ ∂(

h1 ∂ψ)

= −h1h2ω (40)
∂ξ h1 ∂ξ ∂η h2 ∂η


h1h2∂ω

∂t+ ∂ψ

∂η

∂ω

∂ξ− ∂ψ

∂ξ

∂ω

∂η

= ∓ Gr

Re2

(∂h1T

∂η

)+ 1

Re

[∂

∂ξ

(h2

h1

∂ω

∂ξ

)+ ∂

∂η

(h1

h2

∂ω

∂η

)]

(41)

h1h2∂T

∂t+ ∂ψ

∂η

∂T

∂ξ− ∂ψ

∂ξ

∂T

∂η

= 1

Re Pr

[∂

∂ξ

(h2

h1

∂T

∂ξ

)+ ∂

∂η

(h1

h2

∂T

∂η

)](42)

where ξ , η are coordinates in transformed plane and h1, h2are the scale factors given by, h1 = (x2

ξ + y2ξ )1/2 and h2 =

(x2η + y2

η)1/2, for the orthogonal mapping used here. The gov-erning equations, in terms of ψ and ω, are given in Arpaci andLarsen [21] in Cartesian frame. Representation of the same inan orthogonally mapped plane is easily performed (not shownhere) to finally obtain the equations given above. Various non-dimensionalizations remain the same, as in Section 2. In Sec-tion 3, the length scale (L) was chosen as the displacementthickness, a quantity that varied with streamwise distance. Inthis section, a reference length would instead be chosen as aconstant, to be fixed later.

4.1. Boundary and initial conditions

Specification of proper boundary conditions and their imple-mentation into the numerical method is an issue of great impor-tance in DNS. To obtain higher resolution and easily implementboundary conditions, we compute the equations in the orthog-onally transformed (ξ − η)-plane, obtained from the Cartesianframe via the transformations given by,

x(ξ) = xin + LD

[1 − tanh[β1(1 − ξ)]

tanhβ1

](43)

y(η) = Lh

[1 − tanh[β1(1 − η)]

tanhβ1

](44)

with 0 � ξ, η � 1. In the above, xin is the streamwise coordi-nate of the inflow of the computational domain whose stream-wise extent is given by LD = xout − xin, where xout is the loca-tion of the outflow of the computational domain—taken as ei-ther 6L or 12L-depending upon the problem solved. Similarly,the second transformation relates the wall-normal physical dis-tance of Lh to the transformed co-ordinate η. Here, we haveused β1 = 2 to obtain desired grid clustering near the inflowand the wall. A very localized wall-excitation is applied at a lo-cation near the inflow and for this reason the grid clustering inits vicinity is required. This type of tangent hyperbolic transfor-mation apart from producing desired grid clustering, also helpsin reducing aliasing error and is widely used in simulations.

In the above equations, dependent variables represent totalquantities composed of primary and disturbance component. Inthe following, we discuss the boundary conditions separately—so that the process of obtaining the mean and disturbance floware clearly revealed. For the disturbance flow, these boundaryconditions reveal the nature of applied excitation at the wall.However, the boundary conditions for Eqs. (40) to (42) will be

used for the total variable for the receptivity calculations. Atthe inflow and on the top of the domain, free stream boundaryconditions for the primary flow are applied as,

T = 0; ∂ψ

∂η= h2, ω = 0 (45)

Similarly, the required boundary conditions for the primaryflow at the wall are,

T = 1; ψ = constant, ω = − 1

h22

∂2ψ

∂η2(46)

For the disturbance quantities at the wall, we use the no-slipalong with a permeable-wall time-dependent condition, thatrepresents a simultaneous blowing and suction on a localizedstrip to generate waves (as in Fasel and Konzelmann [15]).Corresponding unsteady disturbance flow conditions for non-dimensional variables are given as,

ud = 0, vd = Am sin(βt), Td = 0 (47)

where β is the non-dimensional disturbance frequency and Am

is amplitude of the disturbance. The amplitude function is de-fined along the blowing and suction strip, for x1 � x � xst (asgiven in Fasel and Konzelmann [15]):

Am = 15.1875

(x − x1

xst − x1

)5

− 35.4375

(x − x1

xst − x1

)4

+ 20.25

(x − x1

xst − x1

)3

(48)

And for xst � x � x2:

Am = −15.1875

(x2 − x

x2 − xst

)5

+ 35.4375

(x2 − x

x2 − xst

)4

− 20.25

(x2 − x

x2 − xst

)3

(49)

where xst = (x1 + x2)/2; with x1 and x2 representing the be-ginning and end of the streamwise extent of the strip.

This distribution produces clean localized vorticity distur-bances. The disturbance generation employing blowing andsuction strip, represents spatial downstream development ofdisturbance waves, as originally observed in laboratory experi-ments for flows without heat transfer [16].

At the outflow, boundary conditions for the total variablesare given by,

∂v

∂x= 0; ∂ω

∂t+ Uc

∂ω

∂x= 0

∂T

∂t+ Uc

∂T

∂x= 0 (50)

where Uc is convective speed of the disturbances through theoutflow and here it is taken as the free-stream speed.

For the numerical simulations performed here, results areobtained with a small amplitude of the disturbances at the wallto allow comparison with the results of linear stability the-ory. For the DNS study, we have chosen the parameters asU∞ = 30 m/s, ν = 1.5 × 10−5 m2/s and L = 0.05 m or 0.4 mfor the cases with single and two-loop neutral curves depending


upon the value of Rix . Navier–Stokes equations (40) to (42), to-gether with the boundary conditions specified above are solvedusing high accuracy compact scheme OUCS3, whose detailsare given in Sengupta et al. [22] and [23]. This, formally secondorder accurate scheme, is optimized for least error in the wavenumber plane while evaluating the nonlinear convection termsof the VTE (41). Further details on the basic scheme and bound-ary closures are given in [23]. For the fixed reference Reynoldsnumber (ReL) and Rix , Eqs. (40) to (42) are solved first with theboundary conditions specified for the primary flow to obtain thesteady state solution.

The steady state solution is thereafter perturbed at the wallwith the simultaneous blowing-suction disturbance of the typegiven in Eq. (47) for the fixed frequency parameter Ff =β/ReL. Results are obtained with the amplitude reduced hun-dred times of that used in Fasel and Konzelmann [15] to allowcomparison with the results of linear stability theory. For theBlasius boundary layer excitation that was investigated in [15],the linear stability analysis in Sengupta et al. [20] have shownthe presence of only a few modes (less than four for moderateReynolds numbers), with only one mildly unstable eigenvalueand the rest of them are all stable. In comparison, the mixedconvection opposing flows past vertical plate are more unsta-ble (as shown in Section 3) with significantly higher instabil-ity indicated by the growth rate that is two orders of magni-tude higher. Also, near critical Reynolds number, many stableeigenvalues with similar attenuation rates cause modal inter-actions of the type described in [1,2]. These interactions giverise to additional spatio-temporal excitation—usually at lowerfrequency. These interactions lead to distortion, that may ap-pear as nonlinear—but they are essentially linear in origin. It isfor these reasons, the amplitude of excitation is taken as one-hundredth of the amplitude used in Fasel and Konzelmann [15]to compare with linear theory results, while obtaining percepti-ble response within the chosen computational domain.

To test the numerical method, we compute a case with ReL =105, Rix = 0 and Ff = 1.4 × 10−4, as in Fasel and Konzel-mann [15] with the outflow boundary at xout = 6.0 (correspond-ing Reynolds number based on displacement thickness δ∗, isReδ∗ = 1333) and the location of the top of the computationaldomain is at Lh = 0.35, i.e. y∗ = 32.3δ∗. We have used thetangent hyperbolic grid in streamwise and wall-normal direc-tion with (800 × 300) points. These can be contrasted withthe grid used in Fasel and Konzelmann [15], who took 68 uni-formly distributed points in the wall-normal direction, up to adistance of 6δ∗ only. Disturbances are created at the wall aslocalized blowing and suction from a narrow strip located be-tween x1 = 0.2 and x2 = 0.5, with the corresponding Reynoldsnumber as Reδ∗ = 243.37 and 384.8, respectively—as mea-sured from the leading edge of the plate. For these excitationparameters and location, it is noted from Fig. 3(a) that the dis-turbances first decay before it enters in the region given by theneutral curve and amplify. Inside the neutral loop the distur-bances amplify and they attenuate when they emerge out ofthe neutral curve. The computed disturbance streamwise veloc-ity (ud ) and temperature (Td ), from the solution of Eqs. (40)to (42), are shown in Fig. 5(a) at a time when the unstable flow

pattern is established and does not change further with time. Inall the frames of Fig. 5, we have drawn two vertical lines thatindicate the beginning and end of the unstable region given bythe linear stability analysis for the chosen frequency. These in-formations are obtained by drawing the rays, as in Fig. 4, fromthe origin with the slope indicating the non-dimensional fre-quency Ff . The point of entry of this ray in the neutral looplocate the first vertical line, indicating the beginning of insta-bility for that frequency and the point of exit from the neutralloop indicating the right vertical line. Results show good cor-respondence for the extent of the unstable region obtained bythe linear stability analysis and the present DNS results for thecases with Rix = 0 and 0.0001 shown in Fig. 5(a) and (b). Thisvalidates the numerical method used here for mixed convectionflow the DNS. In Fig. 5, all the frames are shown at t = 30 afterstarting the excitation in each case identically. Thus, the relativereceptivity of the flows are ascertained by comparing the threecases shown in Fig. 5. In Fig. 5(b) and (c), ud and Td are plot-ted as function of x at y∗ = 0.648δ∗ for the opposing flows withRix = 0.0001 and 0.01 for Ff = 1.4 × 10−4, respectively. Onesees slightly higher amplitude for the case in Fig. 5(b) as com-pared to the case in Fig. 5(a), with buoyancy destabilizing theopposing flow. In Fig. 5(c), one notes the envelopes of ud andTd to show significantly higher amplitudes, as compared to thatshown in Fig. 5(a) and (b). There are features of the flow thatshow differences between the results of linear stability analy-sis and DNS for this higher Rix case. Here, the match is onlyqualitative for the extent of the unstable regions as obtained bylinear and nonlinear approaches. The extent is underestimatedby the linear analysis as compared to DNS. In all the cases, ex-citation is applied near the leading edge of the plate, at identicallocations and in its immediate vicinity the attenuating nature ofdisturbance field is not distinctly noticeable.

To provide a quantitative comparison between DNS and lin-ear stability theory, we plot the growth rate as a function ofstreamwise distance for Rix = 0.0 and Ff = 1.4 × 10−4-casein Fig. 6. In the figure, A0 represents the initial amplitude ofdisturbances, taken here at xi = 1.9454 from where DNS indi-cated continuous growth. From the amplitude envelope we havecalculated A(x) to obtain the quantity in the ordinate of the fig-ure. For the linear theory, the corresponding ordinate is obtainedfrom Eq. (12) for the amplitude as,

A(x)

A0(xi)= e

− ∫ xxi

ki dx(51)

Fig. 6 indicates detailed similarity and differences for the DNSand linear stability calculations. Similar comparisons betweenDNS and linear theory were also shown in Seifert and Tumin[24] for different excitation frequencies. They noted differencesincreasing with frequency of excitation.

To test the critical parameters obtained from the linear sta-bility analysis, we compute a case for Ff = 2.79228 × 10−4

(the non-dimensional excitation frequency), that was markedin Fig. 4(a) as OA, touching the critical point, without en-tering the unstable region—for the opposing flow case withRix = 0.01. The solution to Eqs. (40) to (42) are obtained andud is plotted as a function of x in Fig. 7(a), for a fixed height of


Fig. 5. Disturbance quantities ud and Td versus x at y∗ = 0.648δ∗ with Ff = 1.4 × 10−4 excitation. Results are shown after the transients are gone and a timeperiodic state is reached for (a) Rix = 0; (b) opposing flow with Rix = 0.0001 and (c) opposing flow with Rix = 0.01.

Fig. 6. Comparison of growth rates calculated from linear instability theory(LST) and direct numerical simulation (DNS) for the case shown in Fig. 5(a)for Rix = 0.0.

y∗ = 0.648δ∗. The exciter is located at the same location, as inthe case of Fig. 5. Here, as well as in Fig. 5, excitation causesa local solution that is indicated by high amplitude fluctuation,followed by the asymptotic solution—as given by the stabil-ity analysis. From Fig. 7(a), this asymptotic part is seen as the

small amplitude decaying wave solution. As compared to thecase shown in Fig. 5(c), here the disturbance remain all the timein the damped region except when it touches the neutral curveat a single point. This type of continuously decaying solution isseen to occur at the lowest possible frequency—providing thecritical circular frequency. This calculation validates the criticalfrequency obtained from the linear stability results of Table 2.Another interesting phenomenon is also seen from the resultsof DNS reported here. This relates to the spatio-temporal grow-ing wave-front shown in Fig. 7(a). This type of spatio-temporalgrowing wave-front has been reported also in Sengupta et al. [1]for a pure hydrodynamic stability study, where the phenomenonwas established to be related to interactions of multiple de-caying modes for spatially stable systems. Presented results inFig. 7(a) shows that this event is also present for mixed convec-tion flows. From the plotted figures at t = 5 and t = 8, one notesthis leading wave-front to move like a packet in the downstreamdirection, with the amplitude continuously increasing. To showthis phenomenon to be generic, we performed another simula-tion for Rix = 0.01, for which the exciter is at a location beyondthe upper branch of the neutral curve. Corresponding constantphysical frequency line was identified as OB in Fig. 4(a). InFig. 7(b), ud is plotted as a function of x at y∗ = 0.648δ∗for this opposing flow case excited by a non-dimensional fre-


quency, Ff = 1.4 × 10−4. In this case, location of the blow-ing and suction strip is given by x1 = 3.234 (correspondingReδ∗ = 1000) and x2 = 3.5 (corresponding Reδ∗ = 1040.31).

Fig. 7. Disturbance velocity ud versus x at y∗ = 0.648δ∗ for opposing flowwith Rix = 0.01, excited at the wall by frequencies given by (a) Ff =2.79228 × 10−4 and (b) Ff = 1.4 × 10−4.

In this case, following the local solution, the asymptotic part ofthe excitation field is seen also as a decaying wave. In com-parison to the case of Fig. 7(a), here the damping rates aremuch lower and thus, the asymptotic part of the solution is seenover a longer streamwise stretch. The displayed solutions att = 2 and t = 4, once again show the leading spatio-temporallygrowing wave-front. Thus, the spatio-temporal growing wave-front is seen to occur due to interaction of multiple modes forfluid flows—with or without heat transfer. The spatio-temporalwave-front seen here is the same that was described in [1,2]for the Blasius profile. It is not a transient response—as estab-lished in Sengupta et al. [1,2] by solving the problem usingboth linear receptivity approach and solution of full Navier–Stokes equation. In the linear receptivity study, governing Orr–Sommerfeld equation was solved by Bromwich contour integralmethod described in [2,20]. The establishment of the presenceof spatio-temporal growing wave-front is worthwhile, as it canexplain discrepancies between experimental observation andnormal-mode linear stability theory results. We noted in [1,2]and once again here that the leading wave-packet displays vari-ation at lower wave numbers and frequencies as compared tothe asymptotic waves. This has an important significance. Forillustration purpose, let us consider the case where the shearlayer is excited by a constant frequency—as indicated by theline OB in Fig. 4(a). If the center of the asymptotic solution hasreached a streamwise distance indicated by R, then the lead-ing wave-packet would be ahead of this, say at a point S, wherethe flow experiences an excitation corresponding to the time-scale of the wave-front. The point S can be inside the neutralloop, even though the operational point R is outside the neutral

Fig. 8. Disturbance quantities ud and Td versus x at y∗ = 0.68126δ∗ for excitation at Ff = 3.0 × 10−5, for opposing flow with Rix = 0.04.


Fig. 9. Disturbance quantities ud and Td versus x at y∗ = 0.648δ∗ for excitation at Ff = 1.4 × 10−4, for the case of Rix = 0.0 at the indicated times.

loop—as indicated in Fig. 4(a). Thus, the normal mode analysiswould indicate the boundary layer to be stable, while the lead-ing wave-packet would indicate instability. Once excited, theleading wave-packet continues to exist. This is not a transientand the resultant behavior cannot be interpreted by the normal-mode linear instability theory. However, a receptivity approach,as in [1,2,20], based on linearized analysis can do so. Also, anappropriate DNS like the present one can explain the same.

To explain special features of mixed convection bound-ary layers, like the existence of double loop in the neutralcurve for opposing flows, we have computed another case forReL = 8.0 × 105, Rix = 0.04 and Ff = 3.0 × 10−5 with thelocation of outflow boundary at xout = 12.0 where the corre-sponding Reδ∗ = 5798.44. The top of the computational domainis located here at Lh = 0.2, i.e. at y∗ = 27.59δ∗ with 1600points in the streamwise direction and 300 points in the wall-normal direction used for the computation. Here, the blowingand suction strip is located between x1 = 0.2 (correspondingReδ∗ = 748.57) and x2 = 0.3 (corresponding Reδ∗ = 916.81).Numerically obtained ud and Td are plotted against x at y∗ =

0.68126δ∗ in Fig. 8. In this figure, the linearly unstable re-gions have been marked again by vertical lines. The two setsof lines correspond to two loops traversed by a single constantfrequency disturbance—as shown in Fig. 4(b). From Fig. 8,existence of a double-loop for the neutral curve is seen qual-itatively only and the extent of stable and unstable region inthe streamwise direction matches only for the lower loop, whilethe upper loop match is not clear. This is due to the travelingwaves propagating downstream exhibit interaction of the ther-mal mode (upper loop) with the hydrodynamic mode (lowerloop mode). Also, due to the occurrence of multiple wave-packets caused by the dominant hydrodynamic and thermalmodes makes the signal to be multi-periodic—even for the sin-gle frequency excitation—as was the case for Rix = 0 shown inFig. 5(a). Also, the amplitude of variation for Td drastically in-creases with Rix for opposing flow. This shows the buoyancyeffects to severely destabilize opposing flows, as affected sig-nificantly by nonlinear effects.

To resolve the issue of linear and non-linear nature of thedisturbance field as obtained by DNS, we have performed two


Fig. 10. Amplitude envelope of ud versus x for the case of Fig. 9, shown forthree times. Note that the output is scaled by the same ratio, as the input ratio.

additional calculations for Rix = 0.0 case with amplitudes of0.01Am and 0.005Am, with Am as defined in Eqs. (48) and (49).In Fig. 9, the results are compared between the two cases at fourdiscrete times. It is clearly noted that the asymptotic part ofthe solution reduces when the amplitude of the blowing-suctiontype excitation is reduced. To ascertain whether these cases rep-resent linear dynamics or not, in Fig. 10, the amplitude envelopeis compared after the higher amplitude excitation case resultsare scaled by a factor of two—this being the ratio of the in-put to the system. Despite the nonlinear and nonparallel natureof the governing equation, the plotted scaled results in Fig. 10shows linear nature of the response of the system.

Finally, the issue of the spatio-temporal wave front is inves-tigated to establish the correct nature of its genesis. In [1] theanalysis for it was made for parallel shear layer and it was notedthat the wave front originates due to the interaction of multiplemodes and they were clearly visible for spatially stable sys-tems. The leading edge of spatially stable and unstable caseswere exactly at identical locations with identical shape (as inFig. 3 of that reference), it is tempting to conclude that this isdue to the initial transients corresponding to a packet createdwith the maximum growth rate of the parallel shear layer at thatReynolds number. This was further explained in [25] where theleading wave fronts for the two spatially stable cases were an-alyzed for Re = 1000 with (a) β0 = 0.05 and (b) β0 = 0.15.These two frequencies correspond to below and above the neu-tral curve for Re = 1000 and the fast Fourier transform of the

signal at a fixed large time revealed the wavenumbers corre-sponding to the wave fronts to be very different.

To explain this aspect further for actual shear layers includ-ing nonlinear and nonparallel nature of instability, we computedtwo more cases for Blasius boundary layer (Rix = 0.0) with theexciter located between x1 and x2, where the Reynolds num-bers, based on local displacement thickness, are 400 and 450,respectively. Also, the frequencies of excitation are chosen asFf = 2.5 × 10−04 and 3.0 × 10−04. The reason for choosingthese frequencies is that the created disturbances would not goinside the neutral loop—as shown by the two rays OA andOB in Fig. 3(a). Thus, the disturbances are created at loca-tions where the shear layer is sub-critical and the subsequentlythey would not be unstable from linear theory point of view, asthey convect downstream. In spite of this, if a spatio-temporalwave front is created, then the italicized hypothesis given inthe previous paragraph is incorrect. In Fig. 11, the solutions forthese two cases are shown at discrete times and one can clearlysee the presence of leading wave fronts for both the cases. Themain wave-packet is due to the instability of the shear layernear the exciter where the neutral curve is different from thatgiven by the linear instability of the parallel shear layer. Thisdemonstrate the nonparallel effects to be significant at high fre-quencies, as has been postulated earlier in the literature. Theproperties of wave fronts for these two frequencies are alsodifferent-once again repudiating the italicized comment of thelast paragraph.

5. Summary

To study the linear stability of mixed convection flows pastvertical plate, one requires a method to obtain it as a parallelflow. These equations have been obtained using new variablesin Eqs. (4) and (5), with the small buoyancy effect modeledby Boussinesq approximation, given by the term involving thebuoyancy parameter Rix , in the momentum equation. As Rixis small, a regular perturbation method is used to obtain themean flow, shown in Fig. 2. Retaining second order terms donot change the equilibrium flow, whose effects have also beentested here on the stability property. The linear spatial stabilityof the equilibrium flow is investigated here using the compoundmatrix method, for mixed convection flow past vertical plate.Results are shown as neutral curves in Figs. 3 and 4 for assist-ing and opposing flows for different values of Rix . Results forthe assisting flows in Fig. 3, display increasing stability with in-creasing assisting buoyancy—as reported by other researchersalso.

A double loop of the neutral curve has been shown here, forthe first time for mixed convection opposing flows, in the limitof Rix � 0.02. The neutral curve shown in Fig. 4(b), clearlydisplays presence of two distinct lobes of the neutral curve.Presence of two loops give rise to two sets of critical Reynoldsnumbers and circular frequencies. We note that due to couplingof momentum and energy equation, the order of the system in-creases, that increases the number of modes for flows with heattransfer, as compared to only very few modes for flows withoutheat transfer.


Fig. 11. Disturbance quantity ud versus x at y∗ = 0.648δ∗ for excitation at Ff = 2.5 × 10−4 (Shown on left column) and Ff = 3.0 × 10−4 (Shown on the rightcolumn), for Rix = 0.0 at the indicated times. Note the rays OA and OB in Fig. 3 correspond to these cases shown here.

To verify the observations of the linear analysis and to in-vestigate the nonlinear and nonparallel nature of flow stabil-ity, two-dimensional Navier–Stokes and energy equations havebeen solved here by an accurate method, when the boundarylayer is excited by harmonic simultaneous suction and blow-ing at the wall. This kind of receptivity study via solution ofNavier–Stokes equation have been reported for flows withoutheat transfer in Fasel and Konzelmann [15] and Sengupta etal. [1]. While Fasel and Konzelmann [15] have shown the exci-tation of unstable Tollmien–Schlichting waves; in Sengupta etal. [1] this was done for both stable and unstable systems, dis-playing a leading wave-front. The present effort is an attemptto establish the same for mixed convection flows for the firsttime and to verify the features of linear stability studies. Thelinear stability analysis features of the flow are clearly seen inthe DNS results. Such similarities are also noted for the oppos-ing flow case of Rix = 0.0001,0.01, shown in Fig. 5(b) and (c).In Fig. 6, we have compared the growth rate calculated by linearstability theory and by DNS, for the case of Fig. 5(a).

We also establish the presence of spatio-temporal grow-ing wave front for mixed convection flows, extending similarobservation in the study by Sengupta et al. [1,2] for Blasiusboundary layer. Results are shown in Fig. 7, for Rix = 0.01for two frequencies of wall excitation. In the first case, the fre-quency is chosen in such a manner that the constant frequencyline is tangential to the tip of the neutral curve—such that thecreated disturbance is always attenuated. This shows up as aseverely attenuated traveling wave that is preceded by a spatio-temporal growing wave front. This result is shown in Fig. 7(a).In Fig. 7(b), a higher frequency is chosen, but the location of thedisturbance source is where the created waves are attenuatedby a lesser degree, as compared to the case of Fig. 7(a). Herealso, one sees a spatially decaying wave, preceded by the spatio-temporal growing wave-front. In Fig. 8, time evolution of dis-turbance field is shown for the disturbance streamwise velocityand temperature field for opposing flow with Rix = 0.04, forwhich the neutral curve in Fig. 4(b) show two loops of the neu-tral curve. Here, wall excitation is applied before the beginning


of the first unstable loop. Presented results show two stream-wise stretches where the disturbances amplify. The length ofthese stretches match only qualitatively with the linear stabilityresults, while the differences are due to nonlinear and nonparal-lel effects. However, the results shown in Figs. 9 and 10, clearlyindicates that the influence of nonlinearity is rather very small,for the case of Rix = 0.0 for the chosen frequency and ampli-tude levels.

The results shown in Figs. 9 to 11, explain further the natureof the disturbance field computed via DNS, focusing upon thenonlinear and nonparallel nature of the flow instability with andwithout heat transfer.

References

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